The Many Faces of Duality

This
article is based on a special meeting of the Mathematical Programming
Study Group which was held to celebrate the work of Steven Vajda.

Duality
is an important principle in Mathematical Programming. In the opening
talk, Paul Williams (University of Southampton) explored different
forms of duality in various areas of mathematics.

Duality exists as a function between objects X and Y. The fundamental properties of duality are:

reflexivity: the dual of the dual brings you back to where you started;

relations in X are reflected in relations in the dual of X.

Examples of duality include:

the complement in set theory;

negation in logic;

the swapping of faces and vertices in convex polyhedra: the dual of
the cube is the octahedron; the dual of the dodecahedron is the
icosahedron; while the tetrahedron is its own dual;

the swapping of regions and vertices in planar graphs.

In
Linear Programming (LP) we can define a dual problem for every problem
by swapping rows and columns; right hand sides and cost coefficients;
and reversing the signs of the inequalities and the direction of
optimisation. As with other forms of duality, the dual of the dual is
the original problem. However, there is a much stronger property with
LP duality: if the original problem is feasible, then both it and the
dual have optimal solutions with the same objective value. This leads
on to the practical uses of LP duality: to sensitivity analysis, the
marginal value of goods, the economic costs of constraints and the
proof of optimality.

Integer
Programming (IP) presents more difficulties. With an LP problem the
optimal solution to a problem lies at a point where all the constraints
are binding. With an IP problem the optimal solution lies at some
integer solution which in general lies strictly inside the surface of
binding constraints.

Figure 1: Optimal Integer Solution Lies Strictly in Interior

Nonetheless
there are theoretical results for the dual of an IP and these have
economic implications for the allocation of fixed costs, project
selection with limited capital, the tariff structures of public
utilities, etc.

Chairing
the meeting, Ailsa Land (London School of Economics) noted that
everybody using LP in the UK was for many years entirely dependent on
Vajda's books, and indeed was almost certainly taught by him, since he
gave courses and one-off lectures in very many places.

Steven
Vajda spoke next. He noted that there had been numerous occurrences of
coincident but independent discovery in science and mathematics. The
greatest was Leibnitz's and Newton's discovery of the calculus. Fifty
years ago, Dantzig and Kantorovich had separately discovered algorithms
for solving LPs.

With a
worked example, he compared Dantzig's Simplex method with Kantorovich's
method, showing the similarities. Both involve looking at values in
tableaux, checking a property to assess whether one is at the optimum
and, if not, exchanging variables. However, while the optimal Simplex
tableau yields dual values, these are not apparent with Kantorovich's
method.

Jakob
Krarup (DIKU, Copenhagen) explored pairs of problems which were duals
in the sense that they were based on the same data and their optimal
solutions were essentially the same. In the seventeenth century Fermat
had posed this problem: given three points in the plane, find a fourth
point such that the sum of the distances from it to the three points is
minimised. Torricelli (inventor of the barometer) had solved this with
an elegant geometric construction. In the eighteenth century, in The Lady's Diary or Woman's Almanack,
Thomas Moss had posed the problem: given a triangle, circumscribe the
largest possible equilateral triangle. The centre of this triangle is
the Torricelli point which solves Fermat's problem.

Figure 2: Construction of the Torricelli Point

Gautam
Appa (London School of Economics) was concerned with duality in the
residual problems left when all the integer variables in a zero-one MIP
problem had been fixed at integer values. New and old results were
derived by this approach for problems such as where to site an
obnoxious facility and cutting a rectangle to minimise the number of
defects. He showed that the main algorithm used in robust regression
(regression which aims to avoid being driven by spurious outliers) does
not necessarily give locally optimal solutions and that multiple
optimal solutions may exist such that any data point could be an
outlier.

The
first stage in solving an IP problem is to find the continuous optimum,
i.e. the optimum solution in which the decision variables are permitted
to take non-integer values. The difference between this continuous
optimum and the integer optimum is known as the duality gap.
Martin Fieldhouse (Haverly Systems Europe) was considering a particular
class of IP problems known as cutting stock problems. These are
concerned with dividing up wide rolls of material (e.g. paper) into
narrower widths. He asked what was the largest duality gap for cutting
stock problems.

It had
been thought that the largest duality gap was 1. This occurs, for
instance in making widths of 50, 38, 36, 28, 24 and 24 from material
100 wide. The continuous solution is:

He had
found a problem with a duality gap of 1.0333 and this had since been
increased to 1.0666 but it was not known whether this was the largest
or, indeed, whether the duality gap was even finite. The reader is
invited to find trim problems with duality gaps greater than 1.

As a
theoretical physicist, Peter Landsburg (University of Southampton)
found the LP community an optimising group and that reminded him of
entropy maximisation. He therefore considered some properties which
entropy functions may, or may not, possess. Typical among these are
superadditivity (entropy of A+B > entropy of A + entropy of B) and
concavity (as in OR). These were discussed and, as an extra, he used
the laws of thermodynamics to "prove" that the arithmetic mean was
greater than the geometric mean.

Mike
Dempster (University of Essex) explored the extension of Mathematical
Programming duality into Nonlinear Programming and Stochastic
Programming, i.e. where the probabilistic response of the system was
explicitly recognised. There were wide areas of application, including
the management of financial portfolios, forestry, electricity
generation and water resources. In each of these one could take actions
now but the response of the system depended on factors beyond our
control, e.g. financial markets, the weather, consumer actions.

The
approach to solving such problems was to develop sub-problems in many
parallel scenarios, ascribe a probability to each scenario and seek to
find solutions which were good across all scenarios. However, the
number of such scenarios grew exponentially with the number of time
periods and sampling was essential to make the technique practicable.
This led to algorithms based on progressive hedging against future
uncertainty and to the concept of the shadow price of information.

At the
culmination of the meeting Lyn Thomas, the President of the OR Society,
presented Steven Vajda with the Companionship of OR in recognition of
his outstanding contribution over 50 years.

After
the meeting about 20 participants repaired to the Senior Common Room at
the London School of Economics where a fine dinner was held in Steven
Vajda's honour. Thanks go to all the participants and in particular to
Susan Powell for organising the meeting and to Maurice Shutler for
arranging for the use of the conference room at the Monopolies and
Mergers Commission.